There is a very nice paper

Correlated double-proton transfer. I. Theory

Zorka Smedarchina, Willem Siebrand, and Antonio Fernández-Ramos

It considers an incredibly simple potential energy surface to describe double proton transfer.

x_1 is the (dimensionless) position of one proton relative to the middle of its donor and acceptor.

x_2 is the corresponding position for the second proton.

The first term describes a quartic potential with an energy barrier for transfer of the proton between the donor and acceptor.

The dimensionless parameter G describes the extent of correlation or coupling between the two hydrogen bonds. The coupling term is chosen to have the important property that it is symmetric in the two co-ordinates but sensitive to their sign. This is an important difference to earlier [rather nice] work by Benderskii et al. who considered competition between two dimensional quantum tunneling paths [instantons] associated with concerted and sequential transfer.

Three types of Potential Energy surface (PES) can occur, depending on the value of G.

The three cases shown below correspond to

a) 0 < G < 1/2

b) 1/2 < G < 1

c) G > 1.

The coordinates x_a=1/2(x_1-x_2) and x_s=1/2(x_1+x_2). The horizontal line between the left and right minimum corresponds to a

**concerted path.**
For the top PES a

**sequential**proton transfer is possible via one of the two intermediate (INT) states.
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